Algebra of Propositional Logic

Question

How can I rewrite the following propositions in their simplest equivalent forms

i.e. Least atomic propositions

1. $(p \land \neg p) \Rightarrow \neg p$

2. $\neg ((p \land\neg p) \Rightarrow \neg q) $

3. $\neg ((p \land q) \Rightarrow r)$

Thanks

Answer 1

As $p\land\neg p = \bot$:

$$\bot \to \neg p == \top\lor\neg p == \top$$

1. $\top$ (simple closed form)
2. $\bot$
3. $\neg((p\land q)\Rightarrow r)$

Answer 2

The third one can be simplified to $p \land q \land \neg r$:

$$\neg (( p \land q) \rightarrow r) \Leftrightarrow \text{ (Implication)}$$

$$\neg (\neg (p \land q) \lor r) \Leftrightarrow \text{ (DeMorgan)}$$

$$\neg \neg (p \land q) \land \neg r \Leftrightarrow \text{ (Double Negation)}$$

$$(p \land q) \land \neg r \Leftrightarrow \text{ (Association)}$$

$$p \land q \land \neg r$$